cubberly



July 16, 1963 w. E. CUBBERLY, JR 3,097,433

BoREHoLE APPARATUS 5 Sheets-Sheet l Filed Sept. 30, 1960 240 INVENTOR.

BY M] July 16, 1963 w. E. CUBBERLY, JR 3,097,433

BOREHOLE APPARATUS Filed Sept. 50. 1960 /x v /V-m VVG/fer i Caer/y,

NVENTOR.

ATTORNEY July 16, 1963 w. E. CUBBERLY, JR 3,097,433

oREsHoLE4 APPARATUS Filed sept. so, 1960 s sheetssheet s United StatesPatent O ce 3,097,433 BDREHLE APPARATUS Walter E. Cubberly, Jr.,Houston, Tex., assignor to Schlumberger Well Surveying Corporation,Houston, Tex., a corporation of Texas Filed Sept. 30, 1960, Ser. No.59,758 Claims. (Cl. 33-178) This invention relates to apparatus forgauging the size of well bores as well as centralizing well loggingtools .in -a well bore. More particularly, this invention relates toadevice which performs a combined centralizercaliper function and is usedwith well logging apparatus.

Rigid arm calipering devices are well known and generally involve one ormore rigid arms urged outwardly from a tool body by spring action intocontinuous contact with the well bore and include devices for measuringthe radial extension of the rigid arms from the tool body to obtainindications of the diameter of the well bore.

Another type of caliper devices utilizes relatively long, arched springsattached to collars on a tool body wherein one of the collars isslidable along the tool body. The springs are normally preformed orcurved to extend outwardly from the tool body to a larger diameter thanthe diameter of the borehole to be measured so that, when inserted intoya borehole, the arched portions of the springs contact the wall of thewell bore and are forced inwardly. Hence, relative movement between thecollars may be sensed to provide an indication of the borehole diameter.In practice, the length of such spring type calipers is at least sixfeet, so that the springs may operate over a wide range of boreholediameters. An example of this kind of caliper may be found in Patent No.2,712,697.

Spring type centralizers, on the other hand, employ relatively strong,preformed, curved springs xed at their ends to a body member so as toextend outwardly normally to a given diameter which is slightly greaterthan the diameter of the borehole. With this construction, theresistance of the springs to compression provides the requiredcentralizring force in boreholes of a particular diameter, yet permitssliding longitudinal movement of the centralizer through the well bore.

Hence, `spring type calipers and spring type ceutralizers are inappositesince a spring caliper requires a relatively long spring to respond to awide range of borehole diameters and `a spring centralizer requires arelatively strong spring sized to a particular diameter to provide thenecessary forces to centralize. Heretofore, it has been necessary to useboth a centralizing device and ra caliper device when a caliper log isdesired with la centralized tool and the caliper device is one of therigid arm types. This type of caliper device has been favorablyintegrated into wall Contact devices which depend upon an electrode padcontacting the wall of the well bore. Hence, the length and complexityof the tool is not increased substantially since the caliper is operatedconjunctively with the device to place the electrode pad in cont-actwith the well bores. However, 4in certain types of well tools, forexample, a sonic logging tool, wall-engaging pads .are not used. Hence,the complexity and length of the tool would be increased if a rigid armcaliper were added to the tool.

Accordingly, it is an object of the present invention to` provide a newand improved centralizing and well calipering device.

A further object of the present invention is to provide la new andimproved combination centralizing-caliper device for use with a soniclogging tool.

A still further object of the present invention is to provide a new andimproved centralizing-caliper device which is relatively short andcompact in arrangement.

Apparatus, in accordance with the present invention, includes a tubularhousing upon which relatively short,

y 3,097,433 Patented July 16, 1963 bowed, spring arms are mounted. Therespective ends of each spring 'arm are supported for movement relativeto one another. The central portions of the bow springs 'are arranged toextend outwardly of the tool body into contact with the well bore wit-ha substantially constant centralizing force and to deflect, in responseto changes in the well bore diameter, thereby to produce relativemovement between the ends of the spring, the relative movement beingsen-sed by a measuring device. The measuring device develops anelectrical signal representative of the well bore diameter. The bowsprings are preformed to `a semi-elliptical shape or curvature so as tonormally extend outwardly of the tool intermediate of a fully collapsedand 'a fully extended position for the central portions of the springswhile a coil spring is .tensione-d between the ends of the bow springstending to extend the bow springs from their normal position to ra fullyextended position. The relationship between the bow springs and coilspring is such that the centering force of the bow springs is maintainedsubstantially constant over a wide range of borehole diameters.

The novel features of the present invention are set forth withparticularity in the appended claims. The present invention, both as toits organization and manner of operation, together with :further objectsand advantages thereof, may best be understood by way of illustrationand example of certain embodiments when taken in conjunction with theIaccompanying ldrawings in which:

FIG. l is an illustration of Iapparatus embodying the present inventiondisposed in a well bore;

PIG. 2 is a view in cross section of the centralizercaliper as embodiedin the present invention;

FIG. 3 is a view in cross section taken along line 3-3 of FIG. 2;

FIG. 4 is a graphical depiction of the relationship of yforces in thecentralizer-caliper;

FIGS. 5 and 6 lare schematic illustrations of various operatingpositions of the centralizer-caliper;

FIG. 7 is an electrical equalizing circuit for use with thecentralizer-caliper;

FIG. 8 is =a partial view in cross section of another ernbodiment of thecaliper-centralizer; and

FIG. 9 is `a View in cross section taken along line 9-9 of FIG. 8.

Referring now to FIG. l, lnumeral 10 identities a typical sonicapparatus arranged to be suspended by a cable 11 and winch (not shown)in a ll-uid iilled borehole 12. The sonic apparatus 10 includes an upperelectronic cartridge '10a and a lower transducer section 10b whichcontains the sonic transducers. A centralizing and caliper device 13,embodying the present invention, is illustrated as coupled to the lowerend of the transducer section 10b. The sonic apparatus 10 may be, forexample, of the type illustrated in the co-pending application of FrankKokesh, application Serial No. 745,548, tiled .Tune 30, 1958, andassigned to the assignee ofthe present invention.

The electronic cartridge 10a is generally a metal housing and containsthe necessary yelectrical circuitry for the transducer section 10b,while the transducer section 10 includes a stilibut somewhat liexible,toil-filled, tubular mem- -ber 14 in which a transmitter T and lixed,`spaced receivers R1 and R2 are mounted on a low velocity supportingmember 15 such as a stilf coiled spring or Teflon rod. In operation, theelectronic circuitry periodically energizes the transmitter T toperiodically emit pulses of acoustic er1- er-gy which pass through thewell fluid 16 in the well bore 12 to the earth formations. The acousticenergy later successively arrives at the receivers R1 and R2 whichrespectively develop electrical signals in response to the arrival ofacoustic energy and the time interval between the `developed electricalsignals is measured by the electrical 3 circuitry which provides, forexample, a measurement of the velocity of the earth formations. Thevelocity of the formations is, of course,y related to the porosity ofthe earth formations.

Referring now to lFIGS. l-3 in particular, the centralizing and caliperdevice 13 generally includes a rigid tubular member or housing 2i) uponwhich bow springs 24a-c are mounted for relative movement betweenV theirends by means of a pair of tubular spring collars 21 and 22 slidablymounted on the housing. The collars 21, 22 in the preferred embodimentof the invention are arranged to be non-rotatable on the tubular memberand have limited longitudinal movement relative to one another lon thetubular member 20.

The calipering of the well bore 12 is accomplished by the spring arms24a-c which are preformed to a semielliptical curvature and have anessentially rectangular cross section. Preferably three springs areemployed which are secured by their `ends to the spring collars 21 and22 by suitable pin connections and the springs are equidistantly spacedabout the central axis of the tool. The central portions `of the springarms 24a-c engage the bore wall and their spacing is a function of thediameter of the Well bore which is related to the relative spacingbetweenI the ends of the spring arms attached to the collars 21, 22. Therelative spacing `between collars 21 and 22 is sensed |by measuringmeans such as a potentiometer 40 (FIG. 2). The centralizing effect ofthe device 13 in the well bore 12 to develop a substantially constantcentralizing force over a range of borehole diameters is accomplishediby the combined action of spring arms '24a-c and a coil spring 25suitably coupled under tension between the ends of the bow springs 24a-ccoupled to the spring collars 21, 22.

As shown in detail in FIGS. 2 and 3, the tubular body member 26 isprovided with an upper pair of longitudinally extending, Adiametricallyopposed, slots 26 and a lower pair `of longitudinally extending,diametrically opposed, slots 27. Over the upper slots '26, the upperspring collar 21 is slidably received on the tubular mem-ber 20, While atubular support member 30 (FIG. 2) is slidably received within thetubular member 20. The upper spring collar 21 has diametrically opposedopenings 31, 31a sized to receive guide pins 32, 32a, the guide pinsVbeing sized to the width of the slots 26 and secured to the innertubular support member 30 by an'exemplary threaded connection. 'I'helength `of the guide pins 32, 32a is such that they extend from supportmember 30 into the openings 31, 31a of the spring collar 21, and thusthe [guide pins 32, 32a and slots 26 prevent rotation of the collar 21and support member 30 relative to `body 20 while the ends of the pair ofslots 26 limit longitudinal movement of the spring collar 21 and innersupport member 30 relative to the tubular member 20.

Similarly, over the lower slots 27, the lower spring collar 22 isslidably received on the tubular member 20, While an inner cylindricalsupport member 35 is slidably received within the tubular member 20.Also, lower spring collar 22 has diametrically opposed openings 36, 36a,which receive guide pins 37, 37a which are threadedly secured to thecylindrical support member 35. Thus, the lower spring collar 22 andcylindrical support member 35 are arranged on the tubular member 20 forlimited longitudinal movement yet will not rotate relative to thetubular member 2G.

The spacing between the pairs of slots 26 and 27 is such that, whensprings 24a-c are completely flat and parallel to the longitudinal axisof the body 20, guide pins 32, 32a are adjacent `one extremity of slots26, Iwhile guide pins 37, 37a are adjacent a corresponding extremity ofslots 27. Hence, with the springs dat, the collars 21, 22 may ybesimultaneously moved over the length of slots 26, 27.

A linear resistance potentiometer 40 (FIG. 2) is received within thetubular member 20 intermediate of the slots 26 and 27 and includes apotentiometer housing 46a coupled to the tubular support member 30 and asliding potentiometer plunger 4o!) adjustably coupled to the cylindricalsupport member 35. Within the tubular support member 30, a cylindricalclosure member 41 is provided to form one end of a fluid tight chamberfor the potentiometer housing 40a, while a tubular, flexible, bootmember 42 has i-ts ends respectively secured =to the open end of thetubular support member 30 and sliding potentiometer plunger fitlb tocomplete the fluid enclosure chamber for the potentiometer, the fluidchamber being filled with toil. Hence, the potentiometer `40 isprotected from such fluids as may be in the Well bore and is pressurebalanced with respect to the hydrostatic pressure of the uid in the wellbore. The electrical conductors 4-3 of the potentiometer 40 are passedthrough the closure member 41 and Y extend upwardly to be coupled to thecircuitry for sending a signal representative of the diameter of 'thewell bore via the cable conductors (not shown) to conventional indicatormeans 44 (FIG. l) at the surface of the earth.

Each of the spring collars 21, 22 has an onter, annular recess 45, 46(FlG. 2) respectively formed by flanges 47, 48 on the collars and snaprings 49, 50` suitably received in grooves in the collars. Annularspring support members 52, 53 are respectively received in each recess45 and 46 and the three flat, semi-elliptical spring members 24a-c areequidistantly spaced about fthe circumference of the annular supportmembers 52 and 53 and pivotally secured thereto. It will, therefore, beappreciated that inward and outward radial displacement of the centralportions of the springs 24a-c will bring about a corresponding relativedisplacement of the spring ends which are attached to the spring collars21, 22 by means of annular support members 52, 53. Also, equallyapparent is the fact that body member 20 may rotate relative fto thesprings 24a-c; hence, build-up of cable torque is prevented.

Coil spring 25 is coupled to lthe spring collars 21, 22 under tension soas to normally urge the collars towards one another, the spring 25 beingsecured to the respective collars by suitable clamping means. Each ofthe spring members 24a-c is covered with a coating 56 of resilientmaterial such as rubber to reduce the generation of sound noises in theWell bore.

In the operating position of the foregoing described assembly, beforeinsertion into the well bore, the leaf springs 24a-c are fully extendedto their maximum calipering and centralizing diameter by virtue of thetension of coil spring 25. When fthe caliper-centnalizer 13 andapparatus 10 are inserted into the well bore, the leaf springs 24a-c aredeflected inwardly. The relative spacing between the collars 21, 22 ismeasured by the potentiometer 4i) as a function of the well borediameter which, of course, determines the outward extension of thecentral portions of the leaf springs 24a-c. The leaf springs 24a-c andcoil spring 25 in combination develop a substantially constantcentralizing force to support the apparatus .10 centrally in the wellbore. Obviously, in relatively vertical well bores, the centralizing or`supporting force required for the apparatus is negligible. However, allwell bores deviate relative to a true vertical so that a component ofthe weight of apparatus is developed and it is this component force thatthe centralizer force supports.

As shown in FIG. l, when the assembly is inserted in a Well bore,thefriction of the leaf springs 24a-c resists downward movement of ltheassembly; however, this friction force is overcome by the greater Weightof the apparatus y10. Since collars 21 and 22 are slidable on member 20,collar `22 is urged upwardly until guide pins 37, 37a abut the upperends of slots 27, while collar 21 assumes a position relative to slots26 `depending upon the deflection of the leaf springs 24a-c. Hence, ifthe Well bore diameter decreases, the collarY 21 moves upwardly and,conversely, if the well bore diameter increases, the collar 21 movesdownwardly. As noted heretofore, springs 24a-c may be completelydepressed without the ends of the leaf lsprings 24a-c becoming fixedrelative to member 20. Thus, the centralizer-caliper is not subject tothe leaf springs binding in small diameter well bores.

In a like manner, when the apparatus 10 is raised upwardly, collar 21shifts to a lowermost position until guide pins 32, 32a :abut .the lowerends of slots 26, while collar 22 assumes a position relative to slots27 dependent upon the deflection of leaf springs 24a-c. Hence, thecentralizer-caliper is not subject to the leaf `springs binding in smalldiameter well bores While moving upwardly. As the apparatus is passedthrough the well bore, it is free to rotate relative to leaf springs24a-c by virtue of the mounting of annular support members 52, 53 oncollars 21 and 22 so that objectionable cable torque is not developed.

'Ihe central pontions of leaf springs 24a-c which contact the well boredetermine the relative spacing between collars 21 and 22. which ismeasured by potentiometer 40 and recorded by recorder 44 as a functionof depth.

Further, to illustrate the present invention, the particularrelationship between the flat spring larms 24a-c and coil spring 25 willnow be more fully explained with reference to FIGS, 4-6. The leafsprings 24a-c, as shown in FIGS. 5 and 6 and explained heretofore, havean initial, preformed, `semi-elliptical, curvature. In FIG. 5, numeral24 indicates the position rthat springs 24a-c -Would normally assume inthe absence of coil spring 25. The three preformed leaf springs inposition 24 thus extend outwardly so that their central wall contactingportions lie on an imaginary circle 57 (FIG. 6) having a diameter il.'Ilhe curvature of leaf springs 24a-c in position 24 is such that asufficient force in the center of the springs 24a-c will completelydepress the springs to be flat or straight or, more precisely, the fulllength of the springs would lie parallel to the longitudinal axis ofbody 20. When the springs 24a-c are so depressed to a flat positionalong the body 20, as shown by numeral 24", the central contactingportions of springs 24a-c lie on an imaginary circle 58 (FIG. 6) havinga vdiameter d which is the minimum diameter of the centralizer-caliper.When springs 24a-c are forced outwardly to ya greater curvature thanposition 24 to a position as indicated by numberal 24', the centralcontacting portions of springs 24a-c lie in an imaginary circle S9 (FIG.6) having a diameter D which is the maximum diameter to be measured.Thus, it will be appreciated that diameter d is intermediate ofdiameters d and D and the central portion of a spring has an initialdeflection or spacing Y0 from the central portion of a spring at a flatposition 24". It will also be appreciated that fthe overall change indeflection AY of the central portion of the springs 24a-c betweenpositions 24 and 24' is accompanied by an overall change in spacing ALbetween the ends of the springs. As will hereinafter be set forth, thechange in spacing AL is a function of the change in deflection AY. Also,as will become apparent from the `discussion to follow, the preformedcurvature of the leaf springs 24a-c in position 24 in combination withthe coil spring 25 cooperate to obtain a substantially constant-centralizing force.

The mathematical analysis of the relationship between the leaf springs24a-c and coil spring 25 will now be Set forth and following this, anexemplary design will be developed on the basis of the mathematicalrelationships between the springs 24a-c and coil spring 2S.

The mathematical equation for preformed shape or curvature of springs24a-c to permit depression from position `24 to the flat position 24 maybe simply derived by considering only one-half of a single bow or leafspring since a bow or leaf spring is symmetrical about its center.Therefore, the equation for the shape of one-half of a leaf spring maybe derived by cantilever beam formulas.

For example, considering the center of a spring as fixed and the end ofthe spring as free, the equation derived for the initial curvaure ofone-half of a leaf spring is:

eater-eea where the free end of the spring in an initial unloadedcondition is considered as the origin; Y equals the deflection of apoint on the spring due to force P applied to the end of the springnormal to the length of the spring; X equals the distance from theorigin along the spring to the aforesaid point at a deflection Y; Y0equals the maximum deflection of the spring due to the applied force;and l equals one-half of the overall length L of the spring.

Equation l above is derived from the tions for one-half of the spring,i.e.,

d2Y M Px-Elz (2) where M equals the bending moment; P equals the forceapplied to the end of the spring normal to the length of the spring andX is the distance from the origin to any point on the spring; E isYoungs modulus of elasticity; I is the moment of inertia about the axisof bending; and

is the second derivative of the relationship between Y and X. Asimplified way of achieving the proper curvature of the preformed springis to support a normally flat spring at its ends and provide aconcentrated load at its center sufficient to deflect the spring to theselected maximum deflection Y0.

It will be appreciated that the centralizing force exerted by the leafsprings is analogous to the force P applied normal to the length of thesprings. However, in addition to the force P, the coil spring alsoexerts a total axial force on the three springs, which is normal to theforce P. Thus, the force Q of the coil spring applied to one leaf springis equal to moment equa- 3 Therefore, assuming the half-spring, asdefined by Equation l above, to be at the initial preformed curvature,the relationship between the force P and the force Q axially applied tothe end of the spring normal to force P at the normal rest position ofthe spring is derived by the strain energy method for a staticallyindeterminate member. The expression for deflection Sp in the directionof the force P is determined by taking the partial derivative of thestrain energy (taking into consideration both of the forces P and Q)with respect to the force P which gives the following equationEvaluating the Equation 3 on the basis that the spring is constrainedfrom deflecting in the direction of the force P, Sp is set equal toZero. Simplifying the equation and solving for P yields 4 which is theexpression for the relationship between P and Q at the initial position.

general form by replacing Y with'CYo so that where C is aproportionality factor relating the new position of the end of thespring due to the additional force P0 to the initial position of the endof the spring due to force P. Hence, for C=1, the end position is at Y0,the initial position.

Equation 3 can also be rewritten to the general form by the samesubstitution of 1CY0 for Y0 so that l2 6 lsVTE-I[Pz-gono] (s) Thedeflection SI, to the new position is also equatable as By substitutingEquation 7 in Equation 6 and rearranging terms, the expression for theforce P vs. the force Q at any position of the spring may be expressedas To expand the above mathematical analysis to apply to the fulllength, three arm centraliZer-caliper, Equation 8 is multiplied by twofor the full length of the spring and further multiplied by three forthe gross centralizing effort of the device and nally multiplied byone-half to determine the minimum centralizing effort, Fmm of thecentralizer. `It should be noted that in determining the minimumcentralizing effort Fmm, the factor of one-half is based upon a threearm centralizer in its normal logging position in a well bore whereintwo of the arms are positioned at an angle of 60 to the lowermostportion or generatrix of the well bore. Therefore,

Since, in Equation 9, the force Q of the coil spring is equal to Fmin=3and the half length l of a leaf spring equals the overall length Ldivided by two, substituting these values in Equation 9 determines theexpression for the minimum force (Fmm) for the three arm centralizer,which is 72EIY0(1 C +12Y0C L3 5L Equation 10 may be simplified byconsidering the force Fs required to flatten one spring which is minEquation 12 is shown plotted in FIG. 4 for values of Fs and C where o Itwill be appreciated from Equation 12 and FIG. 4 that (a) thecentralizing effort is equal to centralizing force would be a straightline function Where Fs is constant regardless of the well bore diameter.The

value of the coil spring force to give this constant response iscomputed by setting the bracketed terms in Equation 12, i.e.

12 3 gEYOQ 5ml equal to zero and solving for which yields In actualpractice, the centralizer is designed so that an ideal 'Q is obtained atthe maximum diameter of the centralizer. Since the coil spring forcewill increase according to its spring rate as the diameter of the wellbore decreases, the actual centralizing force is not ideally constantbut is substantially constant as indicated by the dashed curve 69 inFIG. 4. It will be noted that curve 60 ultimately arrives at a value of3/2 :Fs when the leaf springs atten.

It will be appreciated from the foregoing and particularly Equation 12that both the leaf springs 24a-c and coil springs 25 contribute to thetotal centralizing force. Rewriting Equation 12 as follows:

It will be seen that the term 3/2 FS (l-C) is the component of force FLof the centralizing force Fmm due to the leaf springs and the term 5 Lis the component of force FC due to the coil spring. The proportionalityconstant C is a function of the diameter of the circle enclosing thecentral portions of springs 24a-c and, hence, the scale of well borediameters is shown in FIG. 4 as Well as the values of C. 'In FIG. 4, theminimum diameter d is equal to 4', the diameter E equal to 10% and thediameter D equal to 14".

From the foregoing, it will also be appreciated that the contributionsof the coil spring and leaf springs to the total centralizing form Fm,are complementary and combine to provide a substantially constantcentralizing force over a wide range of well bore diameters. Thiscomplementary relationship is more clearly shown in FIG. 4, wherein astraight line or linear function 61 illustrates the component force Fcof centralizing force Fmm that the coil spring contributes. The straightline or linear function `62 in FIG. 4 illustrates the component force FLof the total centralizing force the leaf springs contribute. It shouldbe noted, however, in actual practice, that the curve 6@ is the resultof a slight curvature of function 61 due to the spring rate of the coilspring and the ratio of travel of the collars to the bow springdeflection and a slight curvature of function 62 because the value of Ldoes not remain constant.

A practical example of a caliper centralizer design is as follows:

Selected known factors are:

The value for l is based upon a 1/16 x 1 spring stock steel SAE 615 0 orequivalent. Next, a range of diameters for the centraliZer-caliper isselected, for example:

d=4 diameter of springs flat on tool body D=14 maximum diameter ofborehole to be calipered.

The next consideration is the potentiometer 40 and how much travel thepotentiometer plunger 40a should have. A suitable value is, for example,2.31 of movement of the potentiometer plunger 40b relative to thepotentiometer housing 40a. From the range of 4" and 14" diametersselected, it is known that the deflection (Y) of a leaf spring will be 5and this deflection should produce a travel of 2.31 between the ends ofthe springs 24a-c and slidable collars 21, 22. The length L of a leafspring to give a -5 deection with its ends moving from 0-2.31, isdetermined from the formula:

where AL equals the relative travel between the sliding collars andmovement of the spring ends (2.3l); y equals the deection of the springand L equals the length of the spring. Hence, solving Equation l5, thelength L of a spring is equal to 26".

The next consideration is the minimum centralizing force Fmm necessarilyrequired to centralize the weight of the apparatus and this, of course,is dependent upon the weight of the apparatus and the maximum deviationin the Well bore in which the apparatus should be centered. For example,a value of 8.5# centralizing force may be adequate for apparatusweighing 56# in 8# mud in deviated well bores up to 10. From the knownvalue of 8.5# for Fmm, the force FS to depress one spring can bedetermined from Equation l2 where n(5::0 and C=0 so that Substitutingthe values thus far obtained into Equation ll, the dellection Y0 of thecurved spring in the position 24 is calculated to be 3.375. Hence, a bowspring with a lAG" x 1 cross section and length of 26l is preformed tohave a center deflection of 3.375. The springs are thus preformed tothis curvature and, when attached to collars 21 and 22, their ends arespaced 2 radially from the axis of body while the central portionsextend outwardly to the circle 57 with a diameter of 10%".

The coil spring -force necessary to complement the leaf spring force isthen calculated by Equation 13 and is found to be 27.25# Of course, tomaintain the substantially constant centralizing force, the spring rateshould be low, for example lit/1. Using the above figures, thecentralizing force Fmm varies generally from 8.5# at a 4 diameter to amaximum value of 10.2# at an intermediate diameter to the value of 8.5#at a 14 diameter, thereby providing a substantially constantcentralizing force.

In the foregoing description, the potentiometer 40 was noted as linear.Thus, its electrical response is a linear function of the position ofplunger 40h in potentiometer housing 40a. On the other hand, the travelof the potentiometer plunger 40b is related to an exponential functionof the deflection of the springs 24a-c (see Equation 15). Hence,.function forming electrical circuit 65, as shown in FIG. 7, may beemployed to convert the electrical signal into a signal which is alinear lfunction of the deflection of springs 24ac. Circuit 65 includesa resistance R and a resistance r coupled across the input which is theelectrical signal from the potentiometer 40 while the output is takenacross resistance r. The values of resistances R and r are selected in aknown manner so that the quadratic electrical input across R and r isconverted to a linear by varying potential across or directlyproportional to the deflection of springs 24a-c.

Turning now to FIGS. `8 and 9, a modification of the rotatableconnection between tubular member 20' and the ends of bow springs 24a-cis illustrated. Since the connections at the ends of the bow springs aresimilar, the description of only one arrangement |will suice. Tubularmember 20 in this modiiication has, as described heretofore,longitudinally extending slots 27. Cylindrical support member 35 isreceived Within body 20X while a tubular collar 22 is slidably mountedon the body 20. Annular support member 53 is, however, rigidly securedto collar 22. Collar Z2 is provided With access openings 36, 36a and theinner surface of the collar adjacent these openings has an annularrecess 67. Pins 37,` 37a inserted through openings 36, 36a and slots 27are threadedly received by support member 25. The pins` 37, 37a aredimensioned to slidably lit in the slot to prevent rotation of supportmember 35 relative to body 20 and terminate short of the bottom surfaceof the annular recess `67 so that the collar 22 is free to rotaterelative to the body member 20. At the same time, the pins 37, 37a inar1- nular recess 67 are movable longitudinally relative to body 20vwhen the collar 22' slides along the body.

The operation of this modified apparatus is, of course, similar to thepreviously described operation.

While particular embodiments of the present invention have been shownand described, it is apparent that changes and modifications may be madewithout departing from this invention in its broader aspects and,thenefore, the aim in the appended claims is to cover all such changesand modifications as 4fall the true spirit and scope of this invention.

I claim:

1. A well tool for use in a \Well bore including a housing, a pair oftubular collars on said housing wherein at least one of said collars isslidable on said housing permitting relative movement between saidcollars, bow springs pivotally connected to said collars, respectively,having a preformed curvature so that the central portions of saidsprings normally extend outwardly of said housing intermediate of afully collapsed and a fully extended position for said central portionsof said springs, said preformed curvature being symmetrical about amidpoint of each spring, the shape of one-half of each spring having acurvature defined by the formula when considered from an origin at themidpoint of a spring wherein Y0 equals the maximum deflection of thespring due to a given applied force, Y equals the deection of a point onthe spring due to the given applied lforce, X equals the distance fromthe origin along the spring to the aforesaid point at a deflection Y,and l equals one-half the overall length of the spring, and a coiledspring coupled in tension between said collars for extending said bowsprings to a fully extended position when unconstrained by a bore wall.

2. A well tool for use in a Well bore including a housing, a pair oftubular collars on said housing wherein at least one of said collars isslidable on said housing permitting relative movement between saidcollars, bow springs pivotally connected to said collars and having aprefonmed, semi-elliptical curvature so that the ce-ntral portions ofsaid springs normally extend outwardly of said housing intermediate of afully collapsed and a fully `extended position for said central portionsof said springs yet -will assume a at position when fully collapsed, anda coiled spring coupled in tension between :said collars for extendingsaid springs to a fully extended position when unconstrained by a borewall.

3. A well tool Ifor use in a Well bore including a housing, a pair oftubular collars on said housing fwherein at least one of said collars isslidable on said housing, bow springs pivotally connected to saidcollars and having a preformed curvature so that the central portions ofsaid springs normally extend outwardly of said housing intermediate of afully collapsed and a fully extended position for said central portionsof said springs yet will assume a flat position when fully collapsed,and a coiled spring coupled in tension between said collars forextending said springs to a fully extended position fwhen unconstrainedby a bore Wall, said coil spring having a force equal to 1 l where Lequals the length of a bow spring, Yo equals the initial preformeddeflection of a bow spring and Fs equals the force required to flattenone bow spring. V

4. A well tool for use in a well bore including a housing; a pair oftubular collars on said housing wherein at least one collar is slidableon said housing permitting relative movement between said collars; atleast three bow springs coupled to said collars and respectively havinga preformed curvature so that the central portions of said bow springsnormally extend outwardly of said housing intermediate of a fullycollapsed and a fully extended position for said central portions ofsaid bow springs; and a coil spring coupled in tension between saidcollars for extending said bow springs to a fully exten-ded positionwhen unconstrained by a bore wall, said bow springs and coil springproviding aminimum centralizing force Fmg, for the centralizer wherein:

and where FS is the force required to flatten one spring, C is aproportionality factor calculated from movement of the end of springfrom its initial preformed position to another position due to anadditional force, Yo is the initial preformed deflection of the bowspring, L is the length of a bow spring and Q is the force of the coiledspring.

5. A well tool for use in a well bore including a tubular housing, bowsprings having a preformed curvature so that the central portions of thesprings normally extend outwardly of the housing intermediate of a fullycollapsed and fully extended position for the central portions of thesprings, means attaching the ends of said springs to said housing forrelative longitudinal movement therebetween an-d along said housingmeansto limit longitudinal movement of said attaching means along saidhousing, spring means coupled in tension between said attaching meansfor extending said bow springs to a fully extended position whenunconstrained by a bore wall, and measuring means in said tubularhousing coupled to said attaching means to facilitate measurement ofrelative movement between the ends of said bow springs.

6. An elongated apparatus for surveying a well bore and adapted to bepassed through a well bore including a well tool having a tubularhousing, a pair of tubular collars on said housing said collars beingslidable on said tubular housing permitting relative movement betweensaid collars, means on said housing limiting sliding movement of saidcollars thereon, bow springs coupled to said collars and having apreformed curvature so that the central portions of said springsnormally extend outwardly of said housing intermediate of a fullycollapsed and a fully extended position for said central portions ofsaid springs, a coiled spring coupled in tension between said collarsfor extending said bow springs to a fully extended position whenunconstrained by a bore wall, measuring means in said tubular housing,and means coupling said measuring means to said collars to facilitatemeasurement of relative movement between said collars.

. 7. An elongated apparatus -for surveying a well bore and adapted to bepassed through a well bore including a well tool having a tubularhousing, a pair of tubular collars on said housing, at least one of saidcollars being slidable on said housing, bow springs, means coupling saidybow springs to said collars so that said bow springs are free to rotaterelative to said housing, said springs having a preformed curvature sothat the central portions of said springs normally extend outwardly ofsaid housing intermediate of a fully collapsed and a fully extendedposition for said central portions of said springs, spring means coupledin tension between said collars for extending said bow springs to afully extended position when unconstrained by a bore wall, measuringmeans in said tubular housing, and means coupling said collars to saidmeasuring Vmeans including guide pins extending through slots in saidhousing, said guide pins being connected between at least one of saidcollars and said measuring means whereby measurement of the relativelongitudinal movement of said collars is achieved.

8. An elongated apparatus for surveying a well bore and adapted to bepassed through a well bore including a well tool having a tubularhousing, a pair of tubular collars on said housing, at least one of said`collars being slidable on said housing, at least three bow springsequidistantly spaced about said housing, means coupling said bow springsto said collars at such equidistant locations so that said bow springsare free to rotate relative to said housing, said springs having apreformed curvature so that the central portions of said springsnormally extend outwardly of said housing intermediate of a fullycollapsed and a fully extended position for said central portions ofsaid springs, spring means coupled in tension between said collars forextending said bow springs to a `fully extended position, measuringmeans in said housing, and means coupling said collars to said measuringmeans including guide pins extending through slots in said housing, saidguide pins being -connected between at least one of said collars andsaid measuring means whereby measurement of the relative longitudinalmovement of said collars is achieved.

9. An elongated apparatus for surveying a well bore and adapted to bepassed therethrough including: a well tool having a tubular housing, apair of tubular collars slidably mounted on said housing, bow springs,pivotal connecting means -for the respective en-ds of said bow springs,said pivotal means being mounted onsaid collars, said springs having apreformed curvature so that the central portions thereof normally extendoutwardly of said housing intermediate of a fully collapsed and fullyextended position for the said central portions of said spring, springmeans coupled in tension between said collars for extending said bowsprings to a fully extended position when unconstrained by a bore wall,measuring means in said tubular housing, and means coupling said collarsto said measuring means including guide pins extending through slots insaid housing, said guide pins being connected to said collars and saidmeasuring means whereby measurement of the relative longitudinalmovement between said collars is achieved.

10. The'apparatus of claim 4, and further including means pivotallyconnecting said bow springs to said collars.

Culbertson Aug. 22, 1939 Legrand May 26, 1953

1. A WELL TOOL FOR USE IN A WELL BORE INCLUDING A HOUSING, A PAIR OFTUBULAR COLLARS ON SAID HOUSING WHEREIN AT LEAST ONE OF SAID COLLARS ISSLIDABLE ON SAID HOUSING PERMITTING RELATIVE MOVEMENT BETWEEN SAIDCOLLARS, BOWS SPRINGS PIVOTALLY CONNECTED TO SAID COLLARS, RESPECTIVELYHAVING A PEREFORMED CURVATURE SO THAT THE CENTRAL PORTIONS OF SAIDSPRINGS NORMALLY EXTEND OUTWARDLY OF SAID HOUSING INTERMEDIATE OF AFULLY COLLAPSED AND A FULLY EXTENDED POSITION FOR SAID CENTRAL PORTIONSOF SAID SPRINGS SAID PREFORMED CURVATURE BEING SYMMETRICAL ABOUT AMIDPOINT OF EACH SPRING, THE SHAPE OF ONE-HALF OF EACH SPRING HAVING ACURVATURE DEFINED BY THE FORMULA